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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 651 · 40 questions
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How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 10 (c) 14 (d) 8
  2. 2.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 9409 (d) 10609
  3. 3.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 361 (d) 401
  4. 4.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 3625 (c) 4225 (d) 4235
  5. 5.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 12 (d) 13
  6. 6.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 12
  7. 7.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 4 (c) 40 (d) 16
  8. 8.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1936 (d) 1296
  9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6396 (c) 6094 (d) 6084
  12. 12.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 9 (c) 19 (d) 20
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73
  14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 208 (d) 4
  15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  16. 16.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
  17. 17.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
  18. 18.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  19. 19.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
  21. 21.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 1 and 8 (d) 3 and 6
  22. 22.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 125 (c) 147 (d) 213
  23. 23.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 8091 (d) 7579
  24. 24.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
  25. 25.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  27. 27.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
  28. 28.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3596 (d) 3364
  29. 29.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  31. 31.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 5 and 9 (d) 1 and 13
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  33. 33.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
  34. 34.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 45 (d) 63
  35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
  36. 36.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125
  37. 37.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 148 (d) 52
  38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 57 (d) 55
  39. 39.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 16 (c) 17 (d) 10
  40. 40.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625

Answer key

  1. 9
  2. 10609
  3. 441
  4. 4225
  5. 12
  6. 11
  7. 16
  8. 1936
  9. 181
  10. (x + 8)(x + 9)
  11. 6084
  12. 20
  13. 72
  14. 16
  15. (x + 2)(x + 9)
  16. 180
  17. 9
  18. (x + 4)(x + 8)
  19. (x + 3)(x + 4)
  20. 10
  21. 3 and 6
  22. 125
  23. 7569
  24. 5476
  25. (x + 5)(x + 8)
  26. 8 and 9
  27. 1
  28. 3364
  29. (x + 2)(x + 8)
  30. 17
  31. 5 and 9
  32. (x + 6)(x + 9)
  33. 54
  34. 45
  35. 14
  36. 125
  37. 52
  38. 56
  39. 16
  40. 2025

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